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**Langlands correspondence and Bezrukavnikov's equivalence**

Course given at the University of Sydney, first semester 2019.

A course in two parts:

1) an attempt to explain what the Langlands program is about from an arithmetical point of view;

2) affine Hecke algebras, Deligne-Langlands conjecture and Bezrukavnikov's equivalence.

**Extended abstract and bibliography**

**Notes by Anna Romanova**

(Anna's notes also contain a more complete bibliography for part 1 of the course.)

**Part 1**

Below are hand-written notes and some handouts for the course, produced as it progressed.

**Notes from Gus Lehrer's course last
semester on algebraic number theory.**

**Lecture 1: Introduction to reciprocity**

** Number of solutions handout for first lecture.**

**Lecture 2: Review of algebraic number theory**

** Degree 5 solutions handout for second lecture.**

**Lecture 3: Zeta function and L-functions**

** Excerpt from Mazur-Stein giving Riemann's approximations to the zeta function via more and more roots.**

**Lecture 4: (Gus' lecture, notes thanks to Bregje Pauwels) Artin L-functions**

**Lecture 5: (Gus' second lecture, notes thanks to Bregje Pauwels) Brauer's induction theorem**

**Lecture 6: Overview of the Sato-Tate conjecture**

** Example sheet of first 5000 primes for two curves.**

** From slides of a lecture by Ito containing a manuscript of Sato.**

**Lecture 7: Infinite Galois theory, overview of global class field theory (Artin's point of view).**

**Lecture 8: Structure of local Galois groups; local class field theory.**

** Two interesting historical accounts:**

** K. Conrad: History of class field theory.**

** K. Miyake: Takagi's Class Field Theory -- From where? and to where? **

**Lecture 9: Heuristic derivation of local Langlands for ***GL(2)*; basic rep theory of *p*-adic groups.

**Lecture 10: Precise statement of local Langlands for ***GL(2)* for *p* ≠ 2.

**Lecture 11: Why is ***p = 2* special? Spherical representations and Satake isomorphism.

**Lecture 12: Deligne-Langlands conjecture, affine Hecke algebras and Kazhdan-Lusztig equivalence, rough statement of Bezrukavnikov's equivalence.**